prac1C - t = 0 the rear bumper is at-1 0 b(10 Compute the...

18.02 Practice Exam 1 Problem 1. Let P , Q and R be the points at 1 on the x-axis, 2 on the y-axis and 3 on the z-axis, respectively. a) (6) Express--→ QP and--→ QR in terms of ˆ i , ˆ j and ˆ k . b) (9) Find the cosine of the angle P QR . Problem 2. Let P = (1 , 1 , 1), Q = (0 , 3 , 1) and R = (0 , 1 , 4). a) (10) Find the area of the triangle P QR . b) (5) Find the plane through P , Q and R , expressed in the form ax + by + cz = d . c) (5) Is the line through (1 , 2 , 3) and (2 , 2 , 0) parallel to the plane in part (b)? Explain why or why not. Problem 3. A ladybug is climbing on a Volkswagen Bug (= VW). In its starting position, the the surface of the VW is represented by the unit semicircle x 2 + y 2 = 1, y ≥ 0 in the xy-plane. The road is represented as the x-axis. At time t = 0 the ladybug starts at the front bumper, (1 , 0), and walks counterclockwise around the VW at unit speed relative to the VW. At the same time the VW moves to the right at speed 10. a) (15) Find the parametric formula for the trajectory of the ladybug, and ±nd its position when it reaches the rear bumper. (At

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